Question

At the nose of a missile in flight, the pressure and temperature are 5.6 atm and 850°R, respectively. Calculate the density and specific volume.

Answer 1

To calculate the density and specific volume at the nose of the missile, we first converted given values to SI units and used the Ideal Gas Law. The resulting density is approximately 0.00419 kg/m³ and the specific volume is about 238.66 m³/kg.

Step-by-step explanation:

To calculate the density (ρ) and the specific volume (v) of the gas at the nose of the missile, we can use the Ideal Gas Law, which is PV = nRT. Given that the pressure (P) is 5.6 atm and the temperature (T) is 850 degrees Rankine (°R), we need to convert the pressure to pascals (Pa) because the Ideal Gas Constant (R) in SI units is based on meters, kilograms, seconds, and kelvin.

First, we convert the pressure from atm to Pa: P = 5.6 atm x 101325 Pa/atm = 567420 Pa.

Then, to find density, we rearrange the Ideal Gas Law to ρ = P / RT. We assume the molar mass M of air to be roughly that of nitrogen, which is 28.96 g/mol, or 0.02896 kg/mol for calculations. The Ideal Gas constant, R, is 8.314 J/(mol·K).

Now we convert the temperature from °R to Kelvin (K): T = (850 °R) x (5/9) = 472.22 K.

Substituting all values into the density formula we get:

ρ = 567420 Pa / (8.314 J/(mol·K) x 472.22 K) = 567420 Pa / (3924.69 J/mol) ≈ 0.1447 mol/L. Then we multiply by the molar mass to turn moles into mass:

ρ = 0.1447 mol/L x 0.02896 kg/mol ≈ 0.00419 kg/m³.

The specific volume is simply the inverse of density, v = 1/ρ.

v ≈ 1/0.00419 m³/kg ≈ 238.66 m³/kg.

Answer 2

To solve this problem we will apply the definition of the ideal gas equation, where we will clear the density variable. In turn, the specific volume is the inverse of the density, so once the first term has been completed, we will simply proceed to divide it by 1. According to the definition of 1 atmosphere, this is equivalent in the English system to

`1atm = 2116lb/ft^2`

The ideal gas equation said us that,

PV = nRT

Here,

P = pressure

V = Volume

R = Gas ideal constant

T = Temperature

n = Amount of substance (at this case the mass)

Then

`(n)/(V) = (P)/(RT)`

The amount of substance per volume is the density, then

`\rho = (P)/(RT)`

Replacing with our values,

`\rho = (5.6*2116)/(1716*850)`

`\rho = 0.00812slug/ft^3`

Finally the specific volume would be

`v = (1)/(\rho)`

`v = 123ft^3/slug`